Needs["VisualDSolve`"]

\(cont = 
      ContourPlot[x^2 - x\ y + y^2, {x, \(-1\), 1}, {y, \(-1\), 1}, 
        ContourShading -> False, 
        ContourStyle -> {{AbsoluteThickness[2], Blue}}, PlotPoints -> 50, 
        Contours -> 20, Epilog -> {AbsolutePointSize[8], Point[{0, 0}]}, 
        DisplayFunction -> Identity]; 
a = .5; \ 
k = \(-.1\); 

    sols = PhasePlot[{
          \(x'\)[t] == a\ x[t] - y[t] + k\ x[t] \((x[t]^2 + y[t]^2)\), 
          \(y'\)[t] == x[t] - a\ y[t] + k\ y[t] \((x[t]^2 + y[t]^2)\)}, {
          x[t], y[t]}, {t, 0, 25}, {x, \(-1\), 1}, {y, \(-1\), 1}, 
        InitialValues -> {{.9, .9}}, 
        PlotStyle -> {AbsoluteThickness[2], Red}, ShowInitialValues -> True, 
        InitialPointStyle -> {AbsolutePointSize[6], Red}, 
        DisplayFunction -> Identity]; 

    Show[cont, sols, DisplayFunction -> $DisplayFunction]; \)

\(cont = 
      ContourPlot[x^2 - x\ y + y^2, {x, \(-1\), 1}, {y, \(-1\), 1}, 
        ContourLines -> False, ContourShading -> True, PlotPoints -> 50, 
        Contours -> 20, Epilog -> {AbsolutePointSize[8], Point[{0, 0}]}, 
        DisplayFunction -> Identity]; 
a = .5; \ 
k = \(-.1\); 

    sols = PhasePlot[{
          \(x'\)[t] == a\ x[t] - y[t] + k\ x[t] \((x[t]^2 + y[t]^2)\), 
          \(y'\)[t] == x[t] - a\ y[t] + k\ y[t] \((x[t]^2 + y[t]^2)\)}, {
          x[t], y[t]}, {t, 0, 45}, {x, \(-1\), 1}, {y, \(-1\), 1}, 
        InitialValues -> {{.9, .9}}, 
        PlotStyle -> {AbsoluteThickness[2], White}, 
        ShowInitialValues -> True, 
        InitialPointStyle -> {AbsolutePointSize[8], White}, 
        DisplayFunction -> Identity]; 

    Show[cont, sols, DisplayFunction -> $DisplayFunction]; \)